The present chapter introduces vibrations, establishing a robust basis for comprehending this significant facet of physics and engineering. This chapter examines the phenomenon of vibrations in various natural and artificial systems to ascertain its underlying causes. The following part elucidates pivotal historical occurrences in the realm of vibrations, facilitating our comprehension of these phenomena. This chapter’s primary emphasis is the mathematical modelling of systems with a single degree of freedom. The study aims to investigate the complexities associated with these systems, including both damping and non-damping scenarios. This study’s objective is to comprehensively examine harmonic stimulation’s impact on vibrating systems. The primary aim of this study is to elucidate the intricate relationship between external factors and the corresponding reactions of a given system. This chapter encompasses many scholarly materials for anyone interested in furthering their knowledge. This research analyzes vibrating systems with several degrees of freedom, exhibit continuous (distributed) behavior, and display nonlinearity. These disciplines provide a captivating insight into the realm of vibrational dynamics. This chapter establishes a fundamental framework for comprehending vibrations and their many scientific and engineering utilizations.